Block diagram algebra

A block diagram is a picture of an equation. Block-diagram algebra is just the rules for rewriting that picture without changing the equation it represents. Master it and you can collapse any tangled GNC loop into one transfer function.

Core objects

Rule 1 — Blocks in series (cascade)

WHAT: Two blocks one after another.

HOW / derive it: Let input $=X$. After the first block the signal is $G_1X$. Feed that into the second: output $=G_2(G_1X)=G_1G_2\,X$.

Why this step? Multiplication is associative, so order doesn't matter for scalar transfer functions: $G_1G_2=G_2G_1$.

Rule 2 — Blocks in parallel

WHAT: Same input $X$ splits (take-off point) into two blocks whose outputs meet at a summer.

HOW / derive it: Top branch gives $G_1X$, bottom gives $G_2X$. Summer adds: $Y=G_1X\pm G_2X=(G_1\pm G_2)X$.

Rule 3 — The feedback loop (the big one)

WHAT: Forward path $G(s)$, feedback path $H(s)$, negative summer.

HOW / derive from first principles:

1. Error (summer output): $E = R - B$.
2. Output: $Y = G\,E$.
3. Fed-back signal: $B = H\,Y$.

Substitute $B$: $E = R - HY$. Then $Y=GE=G(R-HY)=GR-GHY$. Collect $Y$: $Y+GHY = GR \;\Rightarrow\; Y(1+GH)=GR$.

Rule 4 — Moving blocks (equivalence moves)

These keep $Y/R$ unchanged. Derive each by insisting signals stay equal.

Worked Example 1 — Collapse a loop with a cascade forward path

System: $R \to [\text{summer}, -] \to G_1 \to G_2 \to Y$, feedback $H$.

• Step 1: Series-combine forward path: $G=G_1G_2$. Why? Rule 1 turns two cascaded blocks into one.
• Step 2: Apply feedback formula: $\dfrac{Y}{R}=\dfrac{G_1G_2}{1+G_1G_2H}$. Why? Now it's the standard single-loop form.

Let $G_1=\frac{2}{s}$, $G_2=\frac{1}{s+3}$, $H=1$: $\frac{Y}{R}=\frac{2/[s(s+3)]}{1+2/[s(s+3)]}=\frac{2}{s^2+3s+2}=\frac{2}{(s+1)(s+2)}.$ Why this step? Multiply top & bottom by $s(s+3)$ to clear the fractions — pure algebra, no new physics.

Worked Example 2 — Move a take-off point

Forward: $R\to[-]\to G \to Y$. A take-off after $G$ feeds back through $H$. We want the feedback tapped before $G$ instead.

• Step 1: Currently $B=HY=H\,G\,E$.
• Step 2: If we move the take-off to before $G$, it now reads $E$ (=input to $G$), not $Y=GE$. To keep $B$ the same we must multiply the moved branch by $G$: new feedback block $=GH$. Why? We inserted $G$ to undo the lost multiplication.
• Check: $B=G\cdot H\cdot E$ — identical to before. ✔

Worked Example 3 — Two loops (inner + outer)

Inner loop: forward $G_2$, feedback $H_2$. Outer loop: forward $G_1$ then the inner block, feedback $H_1$.

• Step 1 (reduce inner): $G_{inner}=\dfrac{G_2}{1+G_2H_2}$. Why? Always collapse the innermost loop first — outer rules need a single block there.
• Step 2 (series): forward path $=G_1 G_{inner}=\dfrac{G_1G_2}{1+G_2H_2}$.
• Step 3 (outer loop): $\frac{Y}{R}=\frac{\dfrac{G_1G_2}{1+G_2H_2}}{1+\dfrac{G_1G_2}{1+G_2H_2}H_1}=\frac{G_1G_2}{1+G_2H_2+G_1G_2H_1}.$ Why this step? Multiply top & bottom by $(1+G_2H_2)$ to clear the nested fraction.

Flashcards

Connections

• Transfer functions — the objects living inside each block.
• Laplace transform — why blocks multiply.
• Feedback control loops — application of Rule 3.
• Signal flow graphs & Mason's gain formula — an algebra-free alternative to these moves.
• Op-amp gain — same $G/(1+GH)$ closed-loop math.
• Stability & characteristic equation — the denominator $1+GH=0$ sets the poles.

Concept Map

Hinglish (regional understanding)